Diffusion-limited Reactions on the Cell surface

Manoj GopalakrishnanDept. of Physics, Virginia Tech.

Collaborators :� Uwe C. Tauber, Dept. of Physics, Virginia Tech.� Kimberly Forsten-Williams, Dept. of Chem.Engg.,Virginia Tech.

What the talk is about

� basic Fibroblast Growth Factor (bFGF) stimulates prolifera-tion of several cell types

� Cells possess both high-affinity (R) and low-affinity(HSPG)binding sites to bind bFGF after secretion.

� Binding of bFGF-R stronger in intact cells(HSPG present)than for isolated R.

� Conjecture: A stable triad R-bFGF-HSPG is formed in vivovia surface coupling between bFGF-R and HSPG (and bFGF-HSPG and R ?)

� Is this a plausible scenario, assuming diffusion-limited con-ditions on the cell surface?

� Why is HSPG present in such large numbers on cell surface([HSPG] � 100[R])?

Outline of the talk

� The Cell membrane–lipids, proteins and all that.. dimen-sions, diffusion constants.. effects of the cytoskeleton net-work on mobility of proteins.

� Growth factors and their receptors on cell surface, high andlow affinity receptors

� Experiments with bFGF and its absorption on cell surface,saturation data.

� Mean-field computation of rate constants.

� Smoluchowski theory of surface reactions.

� Monte Carlo simulations

� Summary and open questions

The Cell membrane

� Cell membrane is a two-dimensional fluid, made of phos-pholipid molecules.

� Typical linear dimension of cell 10 ��� .

� Proteins and other objects are also present and diffuse onthe fluid (The Fluid mosaic model).

� Estimated diffusion coefficients � ��� ������ ��������� for phos-pholipids and � ��� �������� ��! ��"� �#�$�%�&� for large proteins.

� Actual diffusion is often slower by a few orders of magnitudedue to presence of anchored proteins forming fences (forphospholipids) or due to obstruction from the cytoskeletonnetwork (for mobile proteins).

Ligands, Receptors and Proteoglycans

' Cellular processes typically require specific molecules atspecific locations.

' For example, in wound healing, growth factors are secretedby cells into the surrounding fluid medium, which are thenabsorbed by cells using specific receptor molecules.

' Often, it is found that there are more than one type of recep-tors present, with different affinities for ligand molecules.

' For example, basic Fibroblast growth factor (bFGF) is boundby a high-affinity receptor (R) and a low-affinity receptor(Heparan Sulfate Proteoglycan -HSPG).

Experiments on bFGF (L) absorption on cell surface

Ref: M. A. Nugent and E. R. Edelman, Biochemistry 31, 8876(1992).

( 0.55nM solution of bFGF (ligand, L) was used for saturationexperiments.

( 0.5 mL solution per )+*-,/. )�021 cells 3 )+*4)5. )�0�6 bFGF percell.

( Estimated 7 )8,�09090 Receptors(R) per cell and 7 ):*-,;.<)�096HSPG (P).

( Binding (R+L 3 L-R and P+L 3 L-P) and dissociation (R-L 3 R+L and P-L 3 P+L ) experiments done for isolated Rand P, and in intact cells.

( Radio-labeled b-FGF was used in binding experiments, andthe amount bound to R or P was determined using a counter.

Experimental results

a. Saturation in intact cell

b. Saturation for isolated R and HSPG

Mean field theory of binding and dissociation= Under experimental conditions, depletion in ligand density neednot be insignificant.

= >�?A@CBED Surface density of free receptors.

= FG?H@IBJD Surface density of free HSPG.

= K9?A@CBED Bulk density of free Ligands.

= L D Height of the ligand column.

Mean-field theory: basic assumptions

= Ligand density K9?H@IB is uniform everywhere in bulk at alltimes.

= At each time @ , K9?A@CB is depleted by an amount proportionalto the number of L molecules absorbed by R (or P) andenhanced by the number released by R (or P).

M K9?A@CBM @ NOLM >�?A@CBM @

M >�?H@IBM @ N P;Q�R K9?A@CBS>�?H@IBUT V RXW >;Y P >�?A@CB[Z= Q R -binding constant, to be found using saturation data.

= V R - dissocation rate, known from experiments.

.

Effective equation for \^]A_C` :a \�]H_I`a _ b ced \�]H_I`gf c h \�]H_I`Ui jlkm\;n

where d b o kp]!q n c ksrt `ui j2k and h bvxwt .

Steady state:

\�]A_zy { `J| \~} b� ]Aj k i o k q n `S��� f� o fk i \ n �� o k ]Aj2k c o k�q�n�`ui \ fn� i (1)

\;n� c �� o k ]Aj k i o k q�n�` (2)

� From experiments, \ }X� ���-� � \;n and � }X� ������� ��n .� Dissociation constants j�k � ����� ���m��� ���and jl� � ��������� ��� ���

.

� By numerical solution, we find ��k b o klq n b �����9��� ��� ���and��� b o �9q n b �����9� � �m��� ���

.

� In intact cells, R-bFGF is much more stable, with j�k � �����9��� ��� ���.

� The stability of bFGF-HSPG in isolation and in intact cells isnot significantly different, j�� � �����9��� ��� ���

.

Problem: Why is R-L complex much more stable in presence ofHSPG?

Conjecture: In cells, R-bFGF combines with HSPG via surfacecoupling to form the triad R-BFGF-HSPG, which is presumablymore stable.

Diffusion-limited reactions

For the reaction � � �   ¡ , the mean-field rate equation is¢¤£¦¥l§©¨¢«ª ¬ ­;® £ ¥�¯ ªI°�£ ¨±¯ ªI°and the solution is

£ ¥l§©¨U¯ ªC°E² ³® ª ´ ªXµ ªH¶· Spatial variation in concentrations of reactants is ignored (cru-cial in low

¢).

· In general, mean-field solution valid only at¢/¸ ¢º¹

.

· For¢ » ¼

and£ ¥ ¯[½ ° ¬ £ ¨ ¯¾½ °

,£ ¥l§©¨ ¯ ªC°<² ¯C¿ ªI°%À�Á , indepen-

dent of ® .

· For£ ¥ ¯¾½ °~¸ £ ¨ ¯¾½ °

,

à � Ä�Å#Æ Ç È ­JÉ ªËÊ�Ì¾Í Î Ï Ð Ñà � Ä�Å#Æ Ç È ­;®ÓÒ £¦Ô ¯¾½ ° ­ £�Õ ¯¾½ °"Öת Î Ï Ø Ñ

Smoluchowski Theory for Diffusion-limited reactions

Ù An approximation for the continuum where reacting moleculeshave a reaction radius Ú .

Ù Reaction rate Û Ü ÝÞÚ�ßCà�á from dimensional analysis.

Ù Consider particle â as static at origin, and compute the fluxof ã thrugh a hypersphere of radius Ú with diffusion con-stant ä9Ý .

Û+åçæ�æèÜ Ýé ê ë ìîí ïÛ åxæ"æ Ü Ýðòñ�ómô Ý êIõ Ú á�ö ë ìîí ä

Û åçæ�æ Ü ÝÞÚ ßCà�á ë ì/÷ ä

Smoluchowski Theory for L-R + P ø L-P-R reactionù Let us consider the reaction L-R +P ú L-P-R.

ù Since ûýüèþ�ÿ �����Óû��Jþ in cells, û�� � �Jþ� û ü;þ .ù So one L-R molecule is surrounded by a sea of P molecules.

Equation for decay of density ��� ������� from Smoluchwoski ap-proximation is

� ��� ������ � � �� ��� ����������

"!$# � �%� �'&)(+*,� .-ù After neglecting the log-correction, the solution is

��� �����0/ ��� ���21$�43 6587:9

; The time scale for conversion of L-R to L-P-R is < =������ �>ù After putting ( ÿ ���@?BA , the time scale turns out to be C��� �D)E 3GF , still much smaller than the time scale of a minute forabsorption from bulk.

ù So, at least for L-R ú L-P-R reaction, the kinetics is governedby bulk absorption, and not by surface diffusion.

ù The reaction L-P+R ú L-P-R is much slower, since [L-P] C [R].

Monte Carlo simulation of surface diffusion: lengthand time scales

H Average linear spacing between HSPG is I J K L�MONQP4R .

H We choose lattice spacing S)T I L�MUNVP4R .

H The corresponding time increment S$WYX Z\[:]+^`_acb .

H For d I LeM�N�f2gihjR a)k�lGm h , S$WnI LeM�N�o seconds.

H We study a square lattice of size p X L�Mrq .H Two time scales in operation: absorption and dissociation

events take place every 6 seconds, while a single MC diffu-sion time step s S$W .

Overview of simulation results

t In steady state, all R is converted to L-P-R u 100% satu-ration! (Also supported by K. E. Forsten, M. Fannon and M.A. Nugent, J. Theor. Biol. 205, 215 (2000).)

t The best results are obtained with a diffusion coefficientv w x�y�zr{�|G}j~ �)���G�G})�, which is

xey��times smaller than the

estimated value.

t In fact, the mobility of membrane proteins is known to be re-duced by the underlying cytoskeleton network by a factor of10-100 (D. M. Leitner, F. Brown and K. R. Wilson, Biophys.J. 78, 125 (2000)).

t It is also possible that our diffusion scheme is too coarse-grained to see the effects of a reaction radius smaller thanthe lattice spacing.

Summary and Outlook

� The surface reactions occur fast because of thelarge number of HSPG compared to R.

� The asymmetry in the concentrations effectivelyovercomes the limitations of low dimensionality inreaction kinetics.

� However, [HSPG] is not too large as to cause ap-preciable depletion in ligand density, which is notdesirable.

� Effects of possible clustering of HSPG has notbeen taken into account.

� A more systematic treatment of diffusion barriersin the simulation would take the model closer toreality.